Topological Genericity and Large Linear Structures in Function Spaces

We prove that the set of integrable functions on the unit circle for which the analogue of Paley’s theorem for $H^1$ fails is residual in $L^1(\mathbb{T})$. Moreover, we establish algebraic genericity and spaceability results in several Hardy-type function spaces under prescribed conditions on Taylor coefficients, extending phenomena considered in \cite{pandis2024some,NestoridisGenericH1}.